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6 The Dealer’s Problem: Finding the Other Side to the Swap

6 The Dealer’s Problem: Finding the Other Side to the Swap

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OTC MARKETS AND SWAPS

295

better rated ﬁrm AA, and is willing to take a lower ﬁxed rate from AA
The corresponding spreads for BBB-type ﬁrms were 48 bps and 54 bps.
This is one way that the swaps dealer adjusts for the alternative credit risks
associated with AA and BBB.
TABLE 8.4

Dealer Swap Schedule for AA-Type Firms

Dealer Pays (BID) the

5-year T-note Rate+50bps (quarterly) in
exchange for LIBOR3,t ′(␻).

5-year T-note rate+52 bps (quarterly) in
exchange for LIBOR3,t ′(␻).

That is, paying fixed and receiving floating
(a short forward position).

That is, paying floating and receiving fixed
(a long forward position).

What we did for BBB applies here and there is nothing really new. The
cash ﬂow diagrams just have different numbers attached to them. We once
again assume that the 5-year T-Note rate is 5.00% and that NP stands for
notional principal.
FIGURE 8.9 Bid Side in a Dealer-Intermediated Swap with AA

Fixed-rate payment at all times
)
tʹ = NP * ( .0550
4
Swaps Dealer

AA
Floating-rate payment at all times
LIBOR3,tʹ(␻))
tʹ = NP * (
4

A similar picture shows the asking side of the swap from the dealer’s point
of view. In this case, AA pays the swaps dealer a ﬁxed-rate payment every
3 months equal to Notional Principal*(.0552/4) and receives a ﬂoating rate
payment at time t′=NP*(LIBOR3,t ′(␻)/4).
FIGURE 8.10 Asked Side in a Dealer-Intermediated Swap with AA

Fixed-rate payment at all times
)
tʹ = NP *( .0552
4
AA

Swaps Dealer
Floating-rate payment at all times
tʹ = NP * (LIBOR43,tʹ(␻))

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TRADING STRUCTURES BASED ON FORWARD CONTRACTS

Now, for whatever reason as we shall soon see, AA wishes to take a sell
position in the swap (pay ﬂoating and receive ﬁxed). Remember that the receiver
of fixed (payer of the floating rate) is said to be selling the swap.
We know from the previous discussion that selling the swap is economically
equivalent to longing a strip of ED futures contracts. Longing because the
participant who buys the underlying 3-month ED time deposit ends up
receiving the ﬁxed rate on those ED time deposits, assuming they bought them
forward at the futures prices. Once again, since the swap reset dates are quarterly
in our example, the long strip would have to have maturity dates that coincide
with the swap’s reset dates.
Why would anyone take a position such as AA in the swap? Just as we did
for BBB, we turn hedging on its head to back out AA’s likely position in the
spot market. Since AA is long an ED futures (forward) strip AA must, if AA
is a hedger, be short something else that is correlated with future LIBOR3,t ′(␻)
rates.
Recall that a short position in a spot commodity is one that either explicitly
or implicitly plans to buy that commodity in the future and therefore worries
about price increases in the underlying commodity. That is the motive for
hedging. If the underlying commodity increases in price (rates decline), then
the corresponding hedging vehicle’s (futures) price will also increase and a
long futures position will recoup some of the losses incurred in the spot market,
subject to basis risk of course.
So we know that AA is short something. Suppose that AA has currently
(at time t=0) issued long-term (5-year) ﬁnancing at its natural rate in the ﬁxedrate market which is 5.30%. See the chart below, Table 8.5, for AA’s respective
borrowing rates in ﬁxed and ﬂoating markets, given its credit risk.
TABLE 8.5

Credit Spreads in the Spot Market for AA-Type Firms

Credit Rating

Fixed rate

Floating rate

AA

5-year T-Note rate+30bp

LIBOR3 flat

Note that AA is a borrower in the spot ﬁxed-rate bond market who has
locked into paying the ﬁxed rate of 5.30% for ﬁve years. This puts AA at a
potential disadvantage if rates decrease across the board.
If that happens, then AA is locked into too high a rate for the remaining
life of the loan. An analogous situation is the homeowner with a ﬁxed-rate
mortgage who sees the ﬁxed rate decline, and therefore wishes to reﬁnance.

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297

If rates do decline, then AA could repurchase the bonds previously issued,
but they would be selling at a higher price than originally issued, creating a
loss for AA. After repurchasing the existing bonds, AA could reﬁnance by
issuing new bonds at lower rates. However, it would be easier for AA to hedge
itself using swaps.
What AA really wants is ﬂoating rate ﬁnancing rather than ﬁxed-rate
ﬁnancing, because ﬂoating-rate ﬁnancing adjusts to interest rate changes. The
question is how to most cost effectively achieve it. Of course, AA can go out
into the ﬂoating-rate market and get quarterly ﬁnancing at ﬂat LIBOR3,t ′(␻)/4.
AA notices, though, that it could issue ﬁxed-rate debt at annualized 5.30%
and simultaneously enter into a swap with the dealer in which AA pays to
the dealer annualized LIBOR3,t ′(␻) ﬂat and receives from the dealer the ﬁxed
rate of 5.50%.
When the coupons become due every quarter, AA uses the dealer’s ﬁxed
payment to pay off the annualized 5.30% and keeps the extra .20% to offset
its ﬂat LIBOR3,t ′ borrowing rate thereby reducing it to LIBOR3,t ′(␻)–.20%.
This transforms AA’s ﬁxed-rate borrowing into ﬂoating rate borrowing at
an annualized savings of 20bp relative to direct ﬂoating rate ﬁnancing. Figure
8.11 illustrates the situation.
FIGURE 8.11 Synthetic Floating-Rate Financing for AA

LIBOR3,tʹ(␻)
4

Swaps Dealer

AA
.0550
4
\$100 MM

.0530
4

Lenders
(Fixed-Rate Market)

Note that the net overall quarterly cost to AA due to this strategy is,

LIBOR 3,t′ (␻ ) ⎛ .0530 .0550 ⎞ LIBOR 3,t′ (␻ ) .002
+⎜

⎟=
4
4 ⎠
4
4
⎝ 4
which is LIBOR3,t ′(␻)–.002 annualized.

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TRADING STRUCTURES BASED ON FORWARD CONTRACTS

Therefore, AA has saved 20 basis points (.002=.2 of a percent) by issuing
ﬂoating-rate debt and swapping ﬁxed for ﬂoating with the swaps dealer. It
has effectively achieved ﬁxed-rate ﬁnancing as well.
Figure 8.12 shows the complete set of cash ﬂows generated by the swap
with both counterparties AA and BBB.
FIGURE 8.12 Full Set of Swap Cash Flows for BBB, AA, and the

Dealer
LIBOR3,tʹ(␻)

LIBOR3,tʹ(␻)

4

AA

4

Swaps Dealer
.0550
4

\$100 MM

BBB
.0554
4

.0530
4

LIBOR3,tʹ(␻)+50bp

\$100 MM

4

Lenders
(Fixed-Rate Market)

Lenders
(Variable-Rate Market)

8.7 ARE SWAPS A ZERO SUM GAME?
We see from Figure 8.12 the net gains to AA, BBB, and to the swaps dealer.
The dealer makes 4 bp on NP annualized because it receives 5.54% annualized
from BBB and pays out 5.50% annualized to AA.
BBB effectively arranges ﬁxed-rate ﬁnancing and pays out 5.54%
(annualized) plus .50%=6.04% which represents a savings of 16 bp over its
natural ﬁxed-rate borrowing rate of 6.20%(annualized).
Finally, AA has arranged ﬂoating-rate ﬁnancing at a cost of LIBOR3,t ′(␻)–
20bp which is 20 bp below what it would naturally pay in the ﬂoating-rate
market. The sum of cost savings to all parties is therefore 4bp+16bp+
20bp=40bp.
There is a nice way of seeing where these 40bps come from. To explain
this will require a few deﬁnitions. Recall the natural rates available to AA and
BBB summarized in Table 8.6.
TABLE 8.6

Credit Spreads for AA and BBB

Credit-Rated Firms

Fixed rate

Floating rate

AA

5.30% (530bps)

LIBOR3

BBB

6.20% (620bps)

LIBOR3+0.5%

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Definitions
1. The Quality Spread in Fixed is
(QSFIX)=Fixed-rate differential between AA and BBB
=620bps–530bps
=90bps.
2. The Quality Spread in Floating is
(QSFLO)=Floating rate differential between AA and BBB
=(LIBOR3+0.5%)–LIBOR3
=50bps.
Then the sum of all gains to all parties is the difference between QSFIX and
QSFLO. In our example, that difference is 90bps–50bps=40bps.
The economic argument behind this result is called comparative advantage
and the process described in our extensive example was known as ‘Arbitraging
the Swaps Market’. As noted, many researchers and practitioners question
whether there is a real arbitrage opportunity here.
8.8 WHY FINANCIAL INSTITUTIONS USE SWAPS
We have skirted by the issue of why parties such as AA and BBB would want
to transform ﬁxed to ﬂoating-rate liabilities, or ﬂoating to ﬁxed-rate liabilities.
That is, why would AA and BBB want to restructure their balance sheets in
the ﬁrst place?
As an example, let’s look at a ﬁnancial intermediary (FI) such as Bank of
America (B of A). Below is a look at what B of A’s partial balance sheet might
look like.
TABLE 8.7

Bank Of America’s Simplified Balance Sheet

Assets

Liabilities

Fixed-rate 30-year mortgages earning
ROA=rA

6-month CDs
with cost of capital=rD

An immediate problem is apparent here and it goes under the title ‘Gap
Management’.
Asset Portfolio

GAP

Liability Portfolio

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TRADING STRUCTURES BASED ON FORWARD CONTRACTS

Bank of America is in a risky position, because it has funded long-term assets
such as 30-year ﬁxed-rate mortgages with short-term liabilities such as 6-month
certiﬁcates of deposit (CD). The latter are ﬂoating-rate securities because B
of A will have to roll them over, reborrowing the principal every 6 months,
most likely on a rate based on LIBOR6. B of A is thus exposed to potentially
increasing ﬁnancing costs.
The principle that attempts to avoid this issue is called the ‘Matching
Principle’ which is to match the duration of your assets with the duration of
Duration has a technical meaning in that it measures the interest rate
sensitivity of a ﬁrm’s asset and liability portfolio. If one could apply the matching
principle perfectly, then the duration of the ﬁrm’s balance sheet would be
zero and the pure interest-rate risk of the portfolio would be neutralized.
Interest-rate swaps provide a low-cost method for a ﬁrm to accomplish this
goal. Note that the relevant measure of the life of an investment is not generally
its stated maturity. Rather, duration is the correct concept which captures the
risk of an interest-rate sensitive investment. Thirty-year ﬁxed-rate mortgages
appear to be very long-term assets but their duration is in the 7–8 year range.
This means that a 7–8 year interest rate swap would be an appropriate hedging
vehicle.
To pursue this example a little more, we ﬁrst focus on the market
participants. The issuer of a mortgage is the homeowner, who is a borrower
in this case. The investor in the mortgage is Bank of America, who is the
lender in the mortgage transaction. B of A expects to earn the spread between
the rate of return on its ﬁxed-rate mortgages and its cost of funds. This can
be written as rA–E(rD(␻)).
What is the primary risk to B of A in this scenario? The risk is that rD(␻)
increases and B of A’s spread narrows. Potential solutions to this gap
management problem include:
a. Sell the ﬁxed-rate mortgages in the secondary mortgage market. To
reduce the risk of these 30-year mortgages, sell them to Freddie Mac or
Fannie Mae. Then buy them back as mortgage-backed securities.
b. Don’t issue ﬁxed-rate mortgages. Use variable-rate mortgages instead.
c. Hedge with strips of Eurodollar futures (ED futures).
d. Lock in the future costs of funding by forward rate agreements (FRAs).
e. Use swaps (Fixed-for-Floating Plain Vanilla Swaps) to transform
ﬂoating-rate costs of funds into ﬁxed-rate costs of funds.

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301

Each of the above solutions has costs and beneﬁts. Instead of describing those
costs and beneﬁts, it is time to turn to the pricing swaps.
8.9 SWAPS PRICING
What does it mean to price a generic swap? The answer is that it means the
same thing as to price a forward contract, which is to determine the
equilibrium forward price. Swaps are forward strips, so pricing generic swaps
is more complicated than pricing an individual forward contract.
Pricing a generic swap means that the swaps market, and therefore the swaps
dealer, has determined the ﬁxed rate that the ﬁxed-rate payer pays in a generic
ﬁxed for ﬂoating-rate swap. This amounts to ‘pricing the swap currently’.
Note that swap pricing includes pricing at origination which we will focus
on. It also includes pricing after origination. Just as a forward contract has zero
value at origination, but assumes positive and negative values to market
participants, so for swaps the swap can assume a positive or negative value as
the underlying spot price changes.
How does the swaps market come up with the base ﬁxed rates to charge
alternative credit risks such as AA and BBB? Recall deﬁnition 13 which is
repeated here.
The Par Swap Rate is the ﬁxed rate at which the swap has a zero present
value at initiation.
Otherwise, there would be an arbitrage opportunity.
We know from our study of forward contracts that the value of a forward
contract is zero at initiation. We also looked at the arbitrage opportunity
available if this were not true. Since a swap is just a strip of forward contracts,
the same valuation must apply to swaps. The details follow.
When a swaps dealer takes the opposite side of a swaps transaction with a
ﬁrm such as BBB who has bought the swap, it ends up receiving fixed and paying
floating periodically. To get a handle on the pricing problem, we can interpret
the swaps dealer’s positions in terms of bonds.
8.9.1 An Example
Consider a 3-year swap. Suppose that LIBOR12,0 is the spot 1-year LIBOR12
at the beginning of year 1. It applies to determine cash ﬂows to be received
at the end of year 1 and is known at t=0.

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TRADING STRUCTURES BASED ON FORWARD CONTRACTS

Also, LIBOR12,1(␻) is 1-year spot LIBOR12 at the beginning of year 2.
Finally, LIBOR12,2(␻) is 1-year spot LIBOR12 at the beginning of year 3. We
will assume that notional principal (NP) is \$100,000,000.
The swap’s ﬂoating payments are,
FLO1=NP*LIBOR12,0,
FLO2=NP*LIBOR12,1(␻),
FLO3=NP*LIBOR12,2(␻).
The swap’s ﬁxed payments are,
FIX1=NP*R,
FIX2=NP*R,
FIX3=NP*R,
where R is the par swap rate.
Note that all the ﬁxed payments are the same: FIX1=FIX2=FIX3=FIX, as is
characteristic of a ﬁxed-for-ﬂoating swap. The swap’s cash ﬂows are illustrated
in Figure 8.13:
FIGURE 8.13 Cash Flows for an Annual Rate Swap from the Dealer’s

Point of View
FLO1 FIX1

t=0

0.5

1.0

FLO2 FIX2

1.5

2.0

FLO3 FIX3

2.5

3.0 = T

But we can write the cash ﬂows of the swap in terms of two bonds, a
ﬂoating-rate bond and a ﬁxed-rate bond. Bonds differ from swaps in that, for
bonds, NP must be repaid at maturity T=3.
Note that NP is a wash in the swap because it gets paid out via the ﬂoatingrate bond and it is received from the ﬁxed-rate bond. Also note that the swap’s
cash ﬂows are equal to the vertical sum of the two bonds’ cash ﬂows, as illustrated
in Figure 8.14.
The swaps dealer has effectively issued (shorted) a ﬂoating-rate bond and
invested in (longed) a ﬁxed-rate bond. Long the ﬁxed-rate bond because he
is receiving the ﬁxed rate from the counterparty, and short the ﬂoating-rate
bond because he is paying LIBOR12,t ′ to the counterparty at each time t′.

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FIGURE 8.14 Decomposing the Swap’s Cash Flows into its Implicit

Bonds
FLO1

t=0

0.5

1.0

FLO2

2.0

1.5

FLO3

2.5

3.0 = T

+
FIX1

t=0

0.5

1.0

FIX2

1.5

2.0

FIX3

2.5

3.0 = T

=
FLO1 + FIX1

t=0

0.5

1.0

FLO2 + FIX2

1.5

2.0

FLO3 + FIX3

2.5

3.0 = T

The conclusion here is that the swap, from the dealer’s point of view, is
economically equivalent to short the floating-rate bond and long the fixed-rate bond.
The next step is to value each of these bonds and thereby to value the swap.
To do so, we need the appropriate discount rates.
8.9.2 Valuation of the Fixed-Rate Bond
In order to value the ﬁxed-rate bond in which the dealer is long, we need
the appropriate discount rates to apply to the bond’s cash ﬂows. We are in
the world of interest-rate swaps which is a LIBOR world.
So we need the current (t=0) spot LIBOR yield curve. It gives the rates
to be applied to zero-coupon Eurobonds for alternative maturities. Assume
that it is as in Table 8.8.
Our long position in the ﬁxed-rate bond can be decomposed as the sum
of three zero-coupon bonds, and one NP repayment bond as indicated in the
multi-level cash ﬂow diagram, Figure 8.15.

304

TRADING STRUCTURES BASED ON FORWARD CONTRACTS

TABLE 8.8

LIBOR Yield Curve (Spot Rates)

Maturity

Zero-Coupon Bond Yields

1 year

6.0%

2 years

6.5%

3 years

7.0%

FIGURE 8.15 The Implicit Fixed-Rate Bond in a Swap, Written in

Terms of Zero-Coupon Bonds
Bond 4 NP
Bond 3 R * NP
Bond 2 R * NP
Bond 1 R * NP

t=0

1.0

2.0

3.0 = T

The LIBOR zero-yield curve says that the appropriate discount rate to
apply to the cash ﬂow from Bond 1 is 6.00%, the appropriate discount rate
to apply to the cash ﬂow from Bond 2 is 6.5%, and the appropriate discount
rate to apply to the cash ﬂows from Bond 3 and from Bond 4 is 7.0%.
This leads to the pricing formulas for the three zero-coupon bonds indicated:
B0,1, B0,2, B0,3, and the notional principal bond NPB0,3. The ﬁrst three bonds
pay exactly the same coupon, R*NP where R is the par swap rate. The fourth
bond represents the return of the principal NP and is therefore denoted as
NPB0,3.
The time at which we have to value these bonds is t=0, as indicated by the
notation. Given this preliminary setup work, it is now easy to value each bond.

B0,1 =

R * NP
1.0600

1.0
1.0600
= R * NP * B0′ ,1
= R * NP *

OTC MARKETS AND SWAPS

B0,2 =

305

R * NP
(1.065)2

1.0
(1.065)2
= R * NP * B0′ ,2
= R * NP *

B0,3 =

R * NP
(1.070)3

1.0
(1.070)3
= R * NP * B0′ ,3
= R * NP *

NP
(1.070)3
= NP * B0′ ,3

NPB0,3 =

where we have written these bond values in terms of B′0,1, B′0,2, and B′0,3 which
are the unit (\$1 payoff) discount bonds maturing at times t ′=1, 2, and 3.
Therefore, the current value of the ﬁxed-rate bond is,

R * NP R * NP R * NP
NP
+
+
+
2
3
1.0600 (1.065) (1.070) (1.070)3
= B0,1 + B0,2 + B0,3 + NPB0,3

B0,Fixed Rate =

= R * NP * B0′ ,1 + R * NP * B0′ ,2 + R * NP * B0′ ,3 + NP * B0′ ,3

8.9.3 Valuation of the Floating-Rate Bond
The ﬂoating-rate bond is described by its cash ﬂow diagram in Figure 8.16.
In order to price such a set of cash ﬂows with random variable payoffs, we
have to take expected values and then discount them appropriately. This means
that, in order to come up with a price for the variable-rate bond, we have to
replace its cash ﬂows by its expected cash ﬂows, where the expectation is with
respect to information currently available.